Möbius transformation and application to coupled stability analysis of cascaded master and slave networks

In this paper, we use the notion of Möbius transformation to develop a chain stability analysis methodology to verify the coupled stability of master-slave teleoperation systems modeled by a cascade of networks. The methodology tracks the transmitted passivity unit disk of the load through the cascade to the user network and provides less restrictive stability conditions compared to having passive or absolutely stable networks. Since the Möbius analysis is decentralized, as opposed to other existing methods such as Llewellyns, the proposed methodology is more suitable for dynamic distributed systems in which a network block in the cascade is modified, added or taken out. We further introduce the notion of “bounded-G absolute stability” to verify the coupled stability for any prescribed range of loads, denoted by circles in the scattering domain. To show the effectiveness of the method numerical analysis and simulation results of a force-position teleoperation system under constant time delay are provided.